Example 1.  Use the method of "data linearization" to find the logistic curve that fits the data for the population of the U.S. for the years 1900-1990.  Fit the curve  [Graphics:Images/LogisticEquationMod_gr_11.gif]  to the census data for the population of the U.S.
        

Date

Populatlion

[Graphics:Images/LogisticEquationMod_gr_12.gif]

76094000

[Graphics:Images/LogisticEquationMod_gr_13.gif]

92407000

[Graphics:Images/LogisticEquationMod_gr_14.gif]

106461000

[Graphics:Images/LogisticEquationMod_gr_15.gif]

123076741

[Graphics:Images/LogisticEquationMod_gr_16.gif]

132122446

[Graphics:Images/LogisticEquationMod_gr_17.gif]

152271417

[Graphics:Images/LogisticEquationMod_gr_18.gif]

180671158

[Graphics:Images/LogisticEquationMod_gr_19.gif]

205052174

[Graphics:Images/LogisticEquationMod_gr_20.gif]

227224681

[Graphics:Images/LogisticEquationMod_gr_21.gif]

249464396

Solution 1.

Enter the data points into a two dimensional array using millions.  Be careful with your typing !

[Graphics:../Images/LogisticEquationMod_gr_22.gif]

[Graphics:../Images/LogisticEquationMod_gr_23.gif]

Next, a limiting population L, or "carrying capacity" must be estimated.  For this data the number L is not too sensitive, but must be larger than the largest ordinate so that the values  [Graphics:../Images/LogisticEquationMod_gr_24.gif]  are not complex numbers. For illustration, we choose  L = 800  million.

 

[Graphics:../Images/LogisticEquationMod_gr_25.gif]
[Graphics:../Images/LogisticEquationMod_gr_26.gif]

Do a series of intermediate computations.

[Graphics:../Images/LogisticEquationMod_gr_27.gif]



[Graphics:../Images/LogisticEquationMod_gr_28.gif]

[Graphics:../Images/LogisticEquationMod_gr_29.gif]
[Graphics:../Images/LogisticEquationMod_gr_30.gif]

[Graphics:../Images/LogisticEquationMod_gr_31.gif]
[Graphics:../Images/LogisticEquationMod_gr_32.gif]
[Graphics:../Images/LogisticEquationMod_gr_33.gif]
[Graphics:../Images/LogisticEquationMod_gr_34.gif]
[Graphics:../Images/LogisticEquationMod_gr_35.gif]


Now glue together the transformed parts to form the pairs  [Graphics:../Images/LogisticEquationMod_gr_36.gif].  

[Graphics:../Images/LogisticEquationMod_gr_37.gif]
[Graphics:../Images/LogisticEquationMod_gr_38.gif]

Now use the Mathematica procedure  Fit  to get the least squares line in XY-space.  Then we shall graph this line in the transformed XY-plane.  

[Graphics:../Images/LogisticEquationMod_gr_39.gif]
[Graphics:../Images/LogisticEquationMod_gr_40.gif]

Plot the least squares line in XY-space.

[Graphics:../Images/LogisticEquationMod_gr_41.gif]


[Graphics:../Images/LogisticEquationMod_gr_42.gif]

[Graphics:../Images/LogisticEquationMod_gr_43.gif]
[Graphics:../Images/LogisticEquationMod_gr_44.gif]

So the coefficients  A  and  B  are located at nodes  (2,1)  and  (1), respectively:

[Graphics:../Images/LogisticEquationMod_gr_45.gif]

[Graphics:../Images/LogisticEquationMod_gr_46.gif]
[Graphics:../Images/LogisticEquationMod_gr_47.gif]

Use  [Graphics:../Images/LogisticEquationMod_gr_48.gif]  and  a = A  to get the coefficients of   [Graphics:../Images/LogisticEquationMod_gr_49.gif]  back in the original  xy-plane.

[Graphics:../Images/LogisticEquationMod_gr_50.gif]

[Graphics:../Images/LogisticEquationMod_gr_51.gif]
[Graphics:../Images/LogisticEquationMod_gr_52.gif]

When we form the function, we must adjust "x" because we shifted the abscissas to the left.  The actual form of the answer is a little different than what we original planned.

[Graphics:../Images/LogisticEquationMod_gr_53.gif]


[Graphics:../Images/LogisticEquationMod_gr_54.gif]

[Graphics:../Images/LogisticEquationMod_gr_55.gif]

Now graph the function  [Graphics:../Images/LogisticEquationMod_gr_56.gif].  

[Graphics:../Images/LogisticEquationMod_gr_57.gif]


[Graphics:../Images/LogisticEquationMod_gr_58.gif]

[Graphics:../Images/LogisticEquationMod_gr_59.gif]

[Graphics:../Images/LogisticEquationMod_gr_60.gif]

Remark.  The data for this example can be obtained from the  U.S. Census Bureau, Historical National Population Estimates:  July 1, 1900 to July 1, 1999.  

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

(c) John H. Mathews 2004